Average Error: 0.4 → 0.3
Time: 41.3s
Precision: 64
\[0.0 \le u1 \le 1 \land 0.0 \le u2 \le 1\]
\[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
↓
\[\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
↓
\left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5
double f(double u1, double u2) {
        double r83708 = 1.0;
        double r83709 = 6.0;
        double r83710 = r83708 / r83709;
        double r83711 = -2.0;
        double r83712 = u1;
        double r83713 = log(r83712);
        double r83714 = r83711 * r83713;
        double r83715 = 0.5;
        double r83716 = pow(r83714, r83715);
        double r83717 = r83710 * r83716;
        double r83718 = 2.0;
        double r83719 = atan2(1.0, 0.0);
        double r83720 = r83718 * r83719;
        double r83721 = u2;
        double r83722 = r83720 * r83721;
        double r83723 = cos(r83722);
        double r83724 = r83717 * r83723;
        double r83725 = r83724 + r83715;
        return r83725;
}
↓
double f(double u1, double u2) {
        double r83726 = 1.0;
        double r83727 = -2.0;
        double r83728 = u1;
        double r83729 = log(r83728);
        double r83730 = r83727 * r83729;
        double r83731 = 0.5;
        double r83732 = pow(r83730, r83731);
        double r83733 = 6.0;
        double r83734 = r83732 / r83733;
        double r83735 = r83726 * r83734;
        double r83736 = 2.0;
        double r83737 = atan2(1.0, 0.0);
        double r83738 = r83736 * r83737;
        double r83739 = u2;
        double r83740 = r83738 * r83739;
        double r83741 = cos(r83740);
        double r83742 = r83735 * r83741;
        double r83743 = r83742 + r83731;
        return r83743;
}

Error

Bits error versus u1

Bits error versus u2

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.4

    \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]
  2. Using strategy rm

  3. Applied div-inv0.4

    \[\leadsto \left(\color{blue}{\left(1 \cdot \frac{1}{6}\right)} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  4. Applied associate-*l*0.4

    \[\leadsto \color{blue}{\left(1 \cdot \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right)\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  5. Simplified0.3

    \[\leadsto \left(1 \cdot \color{blue}{\frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

  6. Final simplification0.3

    \[\leadsto \left(1 \cdot \frac{{\left(-2 \cdot \log u1\right)}^{0.5}}{6}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5\]

Reproduce

herbie shell --seed 2019199 
(FPCore (u1 u2)
  :name "normal distribution"
  :pre (and (<= 0.0 u1 1.0) (<= 0.0 u2 1.0))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 PI) u2))) 0.5))